Folding optimal polygons
by David DureisseixI have looked at various ways of folding the "optimal" pentagon & hexagon (the largest regular hexagon within a square of paper). Additive constraints are a mathematically exact construction, a finite number of operations (no iterative method) and, of course, a folding sequence as simple as possible.
Hexagon
- The corner B comes in B' on the medium vertical line. This allows us to built the intersection F of the fold AE with the diagonal BD. Reverse the model.
- Fold D onto F.
- Resuming the construction to get the optimal hexagon is also easy.
Pentagon
The goal is now to fold a regular pentagon, as large as possible, within a square of paper. In origami geometry, there exists a lot of techniques to fold an approximate pentagon. Much less are concerned with exact pentagon, and only one about optimal pentagon: R. Morassi, The elusive pentagon, in the proceedings of the First International Meeting of Origami Science and Technology, H. Huzita, editor, Ferrara, pp. 27- 37, 1989. The one proposed herein is much simple.- fold AD onto AB where D is the middle of the edge in order
to build C.
is the golden ratio. - bring C on the horizontal mid-crease.
- bisect the complementary angle.
- bisect again and mark the diagonal BE.
- bisect again. I goes on J.
- half way: B goes on J. Unfold.
- & 8 complete the stellated pentagon.
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